Let \(G\)  be a graph and \(\eta \)  an integer such that \(\textnormal {\textsc {ed}}_{\mathcal {G}_F}(G) \le \eta \) .
Let \(S, T \subseteq V(G)\) .
Then one can compute \(\textnormal {\textsc {fvs}}(G)\)  in time \(n^{\mathcal {O}_\eta (1)}\) .
Furthermore, one can determine whether there exists a minimum feedback vertex set in \(G\)  that contains \(S\)  and separates \(T\)  in time \(n^{\mathcal {O}_\eta (1)}\) .

[Proof sketch]
As the approach is standard while working out the details would be tedious, we only sketch the high-level ideas for completeness.
Observe that by Proposition REF .REF , \(G\)  also has treewidth at most \(\eta +1\)  and we can determine a tree decomposition of width \(\eta +1\)  in polynomial time for fixed \(\eta \)  [1]}.
We let \(\left(R, (B_u)_{u \in V(R)}\right)\)  denote such a tree decomposition.
The size of a minimum feedback vertex set can be obtained with dynamic programming over this tree decomposition.
The subproblem for node \(t \in V(R)\)  with \(Z \subseteq B_t\)  and a partition \(\mathcal {P}\)  of \(B_t - Z\)  can be defined as the size of a minimum feedback vertex set \(Y\)  in \(G_t\) , such that
